发表机构
Escola Politécnica University of São Paulo（圣保罗大学理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出PoliVEM框架，通过C++核心和Python接口统一实现虚拟元方法，涵盖多种力学问题，并通过数值实验验证其通用核心，同时指出高次基函数和直接求解器内存是主要限制。
AI 中文摘要
本文介绍了PoliVEM，一个用于计算固体和结构力学中虚拟元方法（VEM）的软件框架。一个C++17计算核心和一个Python接口将一维梁、二维和三维弹性、轴对称弹性、瞬态扩散和有限应变超弹性置于一个通用实现中。该框架将顶点、边、面和单元的未知量存储在一个层次结构中，从公共多项式数据构造能量、应变和$L^2$投影，并在单元层面保留一致性-稳定化分离。核心将网格、材料、单元、组装器和求解器的职责分离，并通过组合将它们结合起来。新公式提供其投影、离散形式和稳定化，同时重用网格表示、未知量编号、边界条件处理、稀疏组装、代数求解器和Python绑定模式。数值基础设施包括多边形和多面体网格输入、高阶实体编号、缓存投影算子、静态凝聚、着色稀疏组装、基于矩阵结构的线性求解器选择、带线搜索和正则化的增量牛顿法，以及显式和隐式时间积分。三个数值实验通过多项式再现、静态凝聚后的恢复、高阶未知量处理、多边形和多面体网格、补片测试、刚度谱以及与独立有限元计算的比较来检验共享实现。结果验证了梁、二维和三维弹性公式的通用核心。它们还确定了各向异性单元上高阶多项式基的缩放和稀疏直接分解所需的内存是当前实现的主要限制。
英文摘要
This work presents PoliVEM, a software framework for the Virtual Element Method (VEM) in computational solid and structural mechanics. A C++17 computational core and a Python interface place one-dimensional beams, two- and three-dimensional elasticity, axisymmetric elasticity, transient diffusion, and finite-strain hyperelasticity in a common implementation. The framework stores vertex, edge, face, and cell degrees of freedom in one hierarchy, constructs the energy, strain, and $L^2$ projections from common polynomial data, and retains the consistency--stabilization split at the element level. The core separates the mesh, material, element, assembler, and solver responsibilities and combines them by composition. A new formulation supplies its projection, discrete form, and stabilization while reusing the mesh representation, degree-of-freedom numbering, boundary-condition treatment, sparse assembly, algebraic solvers, and Python binding pattern. The numerical infrastructure includes polygonal and polyhedral mesh input, higher-order entity numbering, cached projection operators, static condensation, coloured sparse assembly, linear solver selection based on the matrix structure, an incremental Newton method with line search and regularization, and explicit and implicit time integration. Three numerical experiments examine the shared implementation through polynomial reproduction, recovery after static condensation, higher-order degree-of-freedom handling, polygonal and polyhedral meshes, patch tests, stiffness spectra, and comparisons with independent finite element calculations. The results verify the common core for the beam and two- and three-dimensional elasticity formulations. They also identify the scaling of high-order polynomial bases on anisotropic cells and the memory required by sparse direct factorizations as the main limitations of the current implementation.
